The distribution of member stars in the surroundings of an Open Cluster (OC) can shed light on the process of its formation, evolution and dissolution. The analysis of structural parameters of OCs as a function of their age and position in the Galaxy brings constraints on theoretical models of cluster evolution. The Gaia catalogue is very appropriate to find members of OCs at large distance from their centers. We aim at revisiting the membership lists of OCs from the solar vicinity, in particular by extending these membership lists to the peripheral areas thanks to Gaia EDR3. We used the clustering algorithm HDBSCAN on Gaia parallaxes and proper motions to systematically look for members up to 50 pc from the cluster centers. We fitted a King's function on the radial density profile of these clusters and a Gaussian Mixture Model on their two dimensional distribution of members. We also evaluated the degree of mass segregation of the clusters. Our methodology performs well on 389 clusters out of the 467 selected ones. We report the detection of vast coronae around almost all the clusters and the detection of 71 OCs with tidal tails, multiplying by more than four the number of such structures identified. We find the size of the cores to be on average smaller for old clusters than for young ones. Also, the overall size of the clusters seems to slightly increase with age while the fraction of stars in the halo seems to decrease. As expected the mass segregation is more pronounced in the oldest clusters but a clear trend with age is not seen. OCs are more extended than previously expected, regardless of their age. The decrease in the proportion of stars populating the clusters halos highlights the different cluster evaporation processes and the short timescales they need to affect the clusters. Reported parameters all depend on cluster ages but can not be described as single functions of time.